13 EXTENSIONS OF HERMITIAN OPERATORS 437 (c) Deduce from (b) that there exists at least one bounded positive measure v on R with respect to which the powers /" (n ;> 0) are integrable, and which extends the linear form OCF , so that the Pn form a (not necessarily total) orthonormal system in & " (Use Problem 5 of Section 13.20.) All such measures v have the same "moments " each cn is a rational fraction in the aj and bj (1 <iy <£ n — 1) with integer coefficients, and Pn(f) is given as a function of the cn by the formulas (3) of Problem 1. (d) For each of the measures v defined in (c), if ./£ =^ 0 we may write where is a polynomial of degree ^n — 1, and Deduce that, if */£ ^ 0, (4) f I In order that there should be equality in (4) for one value of £ such that Jt, ^ 0 (in which case there is equality for all such £)> it is necessary and sufficient that poly- nomials should be dense in ^cW- (To show that this condition is sufficient, show that it implies that if a function g e <&c(v) is orthogonal to all polynomials, then f (/ - £)- lg(t) dv(t) = 0 for */£ > 0, and use Problem 16 of Section 14.11.) 3. (a) With the hypotheses and notation of Problem 2, consider two sequences (yn\ (zn) of complex numbers satisfying the recurrence relations (1) Ayn = ^-i^-i 4- anyn 4- bnyn+ i, (2) fjizn = bn. lzn.i-}-anzn^rbnzn+l, where A, p are any complex numbers. Show that B- 1 (3) (^-A) X ^2'*=^n-l(^«-l^n-J;«^-l)-^.-l();m-l^m~3;n,^/n-l). fc = m (b) Deduce the following formulas from (a): (4) Pw~i(A)Qn(A) - Pn( (show that the Qn(A) satisfy (1));ear form ap on F such that